E. V. Berestova. Plancherel-Polya inequality for entire functions of exponential type in $L^2(\mathbb{R}^n)$

Let $\mathfrak{M}_{\sigma,n}^p$, $p>0$, be a set of entire functions~$f$ of~$n$ complex variables with exponential type $\sigma=(\sigma_1,\ldots,\sigma_n)$, $\sigma_k>0$, such that their restrictions to~$\mathbb{R}^n$ belong to $L^p(\mathbb{R}^n)$. In 1937 Plancherel and Polya showed that $\sum_{k \in \mathbb{Z}^n}|f(k)|^p \le c_p(\sigma, n) \|f\|^p_{L^p(\mathbb{R}^n)}$ for $f\in \mathfrak{M}_{\sigma,n}^p$, where $c_p(\sigma, n)$ is a finite constant. We study the Plancherel-Polya inequality for~$p=2$. If $0<\sigma_k\le \pi$, then, by the Whittaker-Kotelnikov-Shannon theorem and its generalization to the multidimensional case established by Plancherel and Polya, we have $c_2(\sigma, n)=1$ and any function $f\in \mathfrak{M}_{\sigma,n}^2$ is extremal. In the general case, we prove that $c_2(\sigma, n)=\prod_{k = 1}^{n}\left\lceil \sigma_k/\pi \right\rceil $ and describe the class of extremal functions. We also write the dual problem $\big|\sum_{k \in \mathbb{Z}^n} (g\ast g)(k)\big| \le d_2(\sigma,n) \|g\|_2^2$, $g \in  L^2\left(\Omega\right)$, prove that $c_2(\sigma,n)=d_2(\sigma,n)$, and describe the class of extremal functions.

Keywords: Plancherel-Polya inequality, Paley-Wiener space, entire function of exponential type, Fourier transform.

The paper was received by the Editorial Office on June 23, 2018.

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Ekaterina Vladimirovna Berestova, Ural Federal University, Yekaterinburg, 620990 Russia,
e-mail: e.v.berestova@urfu.ru


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